It seems reasonable to expect that the tonehole should be
representable as a load along a waveguide bore model, thus
creating a loaded two-port junction with two identical bore ports on
either side of the tonehole. From the relations for the loaded
parallel junction (C.101), in the two-port case
with
, and considering pressure waves rather than force
waves, we have

(10.63)

(10.64)

(10.65)

Thus, the loaded two-port junction can be implemented in ``one-filter
form'' as shown in Fig. 9.48 with
(
) and

Comparing with (9.58), we see that the simplified Keefe tonehole
model with the negative series inertance removed (
), is
equivalent to a loaded two-port waveguide junction with
,
i.e., the parallel load impedance is simply the shunt impedance in the
tonehole model.

Each series impedance
in the split-T model of
Fig. 9.43 can be modeled as a series waveguide
junction with a load of
. To see this, set the transmission
matrix parameters in (9.55) to the values
,
, and
from (9.51) to get

(10.66)

where
is the alpha parameter for a series
loaded waveguide junction involving two impedance
waveguides joined
in series with each other and with a load impedance of
, as
can be seen from (C.99). To obtain exactly the loaded series
scattering relations (C.100), we first switch to the more general
convention in which the ``
'' superscript denotes waves traveling into a junction of any number of waveguides. This exchanges ``
'' with ``
''
at port 2 to yield

Finally, we toggle the reference direction of port 2 (the ``current'' arrow
for
on port 2 in Fig. 9.43) so that velocity is
positive flowing into the junction on both ports (which is the
convention used to derive (C.100) and which is typically followed
in circuit theory). This amounts to negating
, giving